Finite Group Databasebeta

← C3×#88676 #13048613

Subgroups — C3×#88676 #13048613

Order 1296.

537 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C2×C3×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic, complemented
3C3×#6235C211yesmaximal, characteristic, complemented
4C3×#638⋊C3C211yesmaximal, characteristic, complemented
5C3×#6200C211yesmaximal, characteristic, complemented
6C3×#6198C211yesmaximal, characteristic, complemented
7C3×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
8C3×#3811—31nomaximal
9C3×D3×#67—31nomaximal
10#88676C311yesmaximal, characteristic, direct product factor
11C22×C3×H(3,𝔽3)C2211yescharacteristic, complemented
12C2×C3×#272C2211yescharacteristic, complemented
13C2×C3×(C3⋊D3)⋊C3—22no—
14C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
16C2×C3×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
17C3×((C3⋊D3)⋊C3)⋊C2—22no—
18C3×#1084C2211yescharacteristic, complemented
19C3×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
20C22×C3×C3⋊D3D311yescharacteristic, complemented
21C22×C32×D3—31no—
22C2×C3×D32—31no—
23C2×C3×D32—31no—
24C3×#638D311yescharacteristic, complemented
25C32×#67—31no—
26C32×#67—31no—
27C32×#67—31no—
28C32×#67—61no—
29C3×#626—31no—
30C3×#626—31no—
31C3×#626—31no—
32C3×#625—31no—
33C3×#624—31no—
34C3×D3×Dic3—31no—
35C22×(C3⋊D3)⋊C3—21no—
36C22×(C3⋊D3)⋊C3C2×C311yescharacteristic, complemented
37C2×((C3⋊D3)⋊C3)⋊C2C2×C311yescharacteristic, complemented
38#6235C2×C311yescharacteristic, complemented
39#638⋊C3—21no—
40#638⋊C3C2×C311yescharacteristic, complemented
41#6200C2×C311yescharacteristic, complemented
42#6198C2×C311yescharacteristic, complemented
43((C3⋊D3)⋊C3)⋊C4C2×C311yescharacteristic, complemented
44C2×C3×H(3,𝔽3)—21no—
45C2×C3×H(3,𝔽3)C2311yescharacteristic
46C3×#272—21no—
47C3×(C3⋊D3)⋊C3—21no—
48C3×(C3⋊D3)⋊C3—21no—
49C3×(C3⋊D3)⋊C3D412yescomplemented
50C2×C3×#67—91no—
51C3×D3×D4—91no—
52C3×D3×D4—91no—
53#3811—31no—
54D3×#67—31no—
55C22×C33—21no—
56C22×C33C2×D311yescharacteristic, complemented
57C22×C33C2×D311yescharacteristic, complemented
58C2×C3×C3⋊D3—22no—
59C2×C3×C3⋊D3C2×D311yescharacteristic
60C2×C3×C3⋊D3C2×D311yescharacteristic, complemented
61C2×C3×C3⋊D3C2×D311yescharacteristic, complemented
62C2×C32×D3—31no—
63C2×C32×D3—31no—
64C2×C32×D3—31no—
65C2×C32×D3—31no—
66C2×C32×D3—31no—
67C2×C32×D3—62no—
68C2×C32×D3—61no—
69C3×D32—62no—
70C3×D32—62no—
71C3×C3⋊Dic3C2×D311yescharacteristic, complemented
72C32×Dic3—31no—
73C32×Dic3—31no—
74C32×Dic3—31no—
75C32×Dic3—61no—
76C22×H(3,𝔽3)—21no—
77C22×H(3,𝔽3)—21no—
78C22×H(3,𝔽3)—41no—
79C22×H(3,𝔽3)C22×C311yescharacteristic, complemented
80C2×#272C22×C311yescharacteristic, complemented
81C2×(C3⋊D3)⋊C3—24no—
82C2×(C3⋊D3)⋊C3—21no—
83C2×(C3⋊D3)⋊C3—21no—
84C2×(C3⋊D3)⋊C3—22no—
85C2×(C3⋊D3)⋊C3—21no—
86C2×(C3⋊D3)⋊C3C22×C311yescharacteristic
87C2×(C3⋊D3)⋊C3C22×C311yescharacteristic, complemented
88C2×(C3⋊D3)⋊C3C22×C311yescharacteristic, complemented
89((C3⋊D3)⋊C3)⋊C2—22no—
90#1084C22×C311yescharacteristic, complemented
91(C3⋊Dic3)⋊C3—21no—
92(C3⋊Dic3)⋊C3C22×C311yescharacteristic, complemented
93C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
94C23×C32—91no—
95C22×C3⋊D3C3×D311yescharacteristic, complemented
96C22×C3×D3—31no—
97C22×C3×D3—31no—
98C22×C3×D3—61no—
99C22×C3×D3—91no—
100C22×C3×D3—91no—
101C22×C3×D3—91no—
102C22×C3×D3—91no—
103C2×D32—31no—
104C2×D32—31no—
105C32×D4—91no—
106C32×D4—91no—
107#638C3×D311yescharacteristic, complemented
108C3×#67—31no—
109C3×#67—31no—
110C3×#67—31no—
111C3×#67—31no—
112C3×#67—31no—
113C3×#67—31no—
114C3×#67—61no—
115C3×#67—61no—
116C3×#67—61no—
117C3×#67—61no—
118C3×#67—61no—
119C3×#67—62no—
120C3×#67—91no—
121C3×#67—91no—
122C3×#67—91no—
123C3×#67—91no—
124C3×#67—91no—
125C2×C3×Dic3—91no—
126C3×D12—91no—
127C3×D12—91no—
128C3×C4×D3—91no—
129C3×C4×D3—91no—
130#626—31no—
131#626—31no—
132#626—31no—
133#625—31no—
134#624—31no—
135D3×Dic3—31no—
136C2×C33—21no—
137C2×C33—21no—
138C2×C33—21no—
139C2×C33—41no—
140C2×C33C22×D311yescharacteristic
141C2×C33C22×D311yescharacteristic
142C3×C3⋊D3—21no—
143C3×C3⋊D3—21no—
144C3×C3⋊D3#6712yescomplemented
145C32×D3—32no—
146C32×D3—61no—
147C32×D3—61no—
148C32×D3—61no—
149C32×D3—61no—
150C32×D3—121no—
151C2×H(3,𝔽3)—22no—
152C2×H(3,𝔽3)—21no—
153C2×H(3,𝔽3)—21no—
154C2×H(3,𝔽3)—21no—
155C2×H(3,𝔽3)—41no—
156C2×H(3,𝔽3)—41no—
157C2×H(3,𝔽3)—42no—
158C2×H(3,𝔽3)C23×C311yescharacteristic
159#272—21no—
160(C3⋊D3)⋊C3—22no—
161(C3⋊D3)⋊C3—21no—
162(C3⋊D3)⋊C3—21no—
163(C3⋊D3)⋊C3—22no—
164(C3⋊D3)⋊C3—41no—
165(C3⋊D3)⋊C3C3×D412yescomplemented
166C2×C3×D4—271no—
167C2×#67—91no—
168D3×D4—91no—
169D3×D4—91no—
170C22×C32—21no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—31no—
174C22×C32—31no—
175C22×C32—41no—
176C22×C32—61no—
177C22×C32—61no—
178C22×C32—62no—
179C22×C32—61no—
180C22×C32—91no—
181C22×C32—91no—
182C22×C32—91no—
183C22×C32—91no—
184C22×C32—182no—
185C22×C32D3211yescharacteristic
186C22×C32C2×C3×D311yescharacteristic, complemented
187C22×C32C2×C3×D311yescharacteristic, complemented
188C2×C3⋊D3—22no—
189C2×C3⋊D3C2×C3×D311yescharacteristic
190C2×C3⋊D3C2×C3×D311yescharacteristic, complemented
191C2×C3⋊D3C2×C3×D311yescharacteristic, complemented
192C2×C3×D3—31no—
193C2×C3×D3—31no—
194C2×C3×D3—31no—
195C2×C3×D3—31no—
196C2×C3×D3—31no—
197C2×C3×D3—31no—
198C2×C3×D3—31no—
199C2×C3×D3—31no—
200C2×C3×D3—31no—
201C2×C3×D3—31no—
202C2×C3×D3—61no—
203C2×C3×D3—61no—
204C2×C3×D3—61no—
205C2×C3×D3—61no—
206C2×C3×D3—61no—
207C2×C3×D3—61no—
208C2×C3×D3—61no—
209C2×C3×D3—62no—
210C2×C3×D3—62no—
211C2×C3×D3—64no—
212C2×C3×D3—62no—
213C2×C3×D3—91no—
214C2×C3×D3—91no—
215C2×C3×D3—91no—
216C2×C3×D3—182no—
217C2×C3×D3—182no—
218C2×C3×D3—182no—
219C2×C3×D3—182no—
220D32—62no—
221D32—62no—
222C32×C4—91no—
223C32×C4—91no—
224C3⋊Dic3C2×C3×D311yescharacteristic, complemented
225C3×Dic3—31no—
226C3×Dic3—31no—
227C3×Dic3—31no—
228C3×Dic3—31no—
229C3×Dic3—31no—
230C3×Dic3—31no—
231C3×Dic3—61no—
232C3×Dic3—61no—
233C3×Dic3—61no—
234C3×Dic3—61no—
235C3×Dic3—61no—
236C3×Dic3—62no—
237C3×Dic3—91no—
238C33—21no—
239C33D3×D411yescharacteristic, complemented
240C33C2×#6711yescharacteristic, complemented
241H(3,𝔽3)—21no—
242H(3,𝔽3)—21no—
243H(3,𝔽3)—41no—
244H(3,𝔽3)C2×C3×D411yescharacteristic, complemented
245C23×C3—91no—
246C23×C3—91no—
247C23×C3—181no—
248C23×C3—271no—
249C22×D3—31no—
250C22×D3—91no—
251C22×D3—91no—
252C22×D3—91no—
253C22×D3—91no—
254C3×D4—91no—
255C3×D4—91no—
256C3×D4—91no—
257C3×D4—91no—
258C3×D4—181no—
259C3×D4—181no—
260C3×D4—271no—
261C3×D4—271no—
262C2×C3×C4—271no—
263#67—31no—
264#67—31no—
265#67—31no—
266#67—61no—
267#67—91no—
268#67—91no—
269#67—91no—
270#67—91no—
271#67—91no—
272C2×Dic3—91no—
273D12—91no—
274D12—91no—
275C4×D3—91no—
276C4×D3—91no—
277C2×C32—21no—
278C2×C32—22no—
279C2×C32—21no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—31no—
285C2×C32—31no—
286C2×C32—41no—
287C2×C32—41no—
288C2×C32—42no—
289C2×C32—41no—
290C2×C32—61no—
291C2×C32—61no—
292C2×C32—61no—
293C2×C32—61no—
294C2×C32—62no—
295C2×C32—61no—
296C2×C32—92no—
297C2×C32—121no—
298C2×C32—121no—
299C2×C32—121no—
300C2×C32—122no—
301C2×C32—181no—
302C2×C32—181no—
303C2×C32—181no—
304C2×C32C2×D3211yescharacteristic
305C2×C32C22×C3×D311yescharacteristic
306C2×C32C22×C3×D311yescharacteristic
307C3⋊D3—21no—
308C3⋊D3—21no—
309C3⋊D3C3×#6712yescomplemented
310C3×D3—32no—
311C3×D3—32no—
312C3×D3—62no—
313C3×D3—62no—
314C3×D3—61no—
315C3×D3—61no—
316C3×D3—61no—
317C3×D3—61no—
318C3×D3—61no—
319C3×D3—61no—
320C3×D3—61no—
321C3×D3—61no—
322C3×D3—92no—
323C3×D3—121no—
324C3×D3—121no—
325C3×D3—121no—
326C3×D3—121no—
327C3×D3—121no—
328C3×D3—122no—
329C3×D3—181no—
330C3×D3—181no—
331C2×D4—271noSylow
332C22×C3—21no—
333C22×C3—31no—
334C22×C3—31no—
335C22×C3—61no—
336C22×C3—61no—
337C22×C3—61no—
338C22×C3—91no—
339C22×C3—91no—
340C22×C3—91no—
341C22×C3—91no—
342C22×C3—91no—
343C22×C3—91no—
344C22×C3—91no—
345C22×C3—91no—
346C22×C3—121no—
347C22×C3—181no—
348C22×C3—181no—
349C22×C3—181no—
350C22×C3—181no—
351C22×C3—182no—
352C22×C3—182no—
353C22×C3—184no—
354C22×C3—542no—
355C22×C3((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
356C22×C3C3×D3211yescharacteristic
357C2×D3—31no—
358C2×D3—31no—
359C2×D3—31no—
360C2×D3—31no—
361C2×D3—31no—
362C2×D3—61no—
363C2×D3—62no—
364C2×D3—91no—
365C2×D3—91no—
366C2×D3—91no—
367C2×D3—182no—
368C2×D3—182no—
369C2×D3—182no—
370C2×D3—182no—
371C3×C4—91no—
372C3×C4—91no—
373C3×C4—91no—
374C3×C4—91no—
375C3×C4—181no—
376C3×C4—181no—
377Dic3—31no—
378Dic3—31no—
379Dic3—31no—
380Dic3—61no—
381Dic3—91no—
382C32—21no—
383C32—21no—
384C32—21no—
385C32—31no—
386C32—31no—
387C32—41no—
388C32—61no—
389C32—61no—
390C32—62no—
391C32—61no—
392C32D3×#6711yescharacteristic
393C32C3×D3×D411yescharacteristic, complemented
394C32C2×C3×#6711yescharacteristic, complemented
395C23—91no—
396C23—271no—
397D4—91no—
398D4—91no—
399D4—271no—
400D4—271no—
401C2×C4—271no—
402C2×C3—21no—
403C2×C3—21no—
404C2×C3—21no—
405C2×C3—31no—
406C2×C3—31no—
407C2×C3—41no—
408C2×C3—61no—
409C2×C3—61no—
410C2×C3—62no—
411C2×C3—61no—
412C2×C3—61no—
413C2×C3—61no—
414C2×C3—92no—
415C2×C3—92no—
416C2×C3—121no—
417C2×C3—121no—
418C2×C3—122no—
419C2×C3—121no—
420C2×C3—182no—
421C2×C3—181no—
422C2×C3—181no—
423C2×C3—182no—
424C2×C3—181no—
425C2×C3—181no—
426C2×C3—181no—
427C2×C3—181no—
428C2×C3—361no—
429C2×C3—361no—
430C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
431C2×C3C2×C3×D3211yescharacteristic
432D3—32no—
433D3—61no—
434D3—61no—
435D3—61no—
436D3—61no—
437D3—92no—
438D3—121no—
439D3—181no—
440D3—181no—
441C22—91no—
442C22—91no—
443C22—91no—
444C22—91no—
445C22—182no—
446C22—542no—
447C22C3×((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
448C4—91no—
449C4—91no—
450C3—21no—
451C3—31no—
452C3—31no—
453C3—61no—
454C3—61no—
455C3—61no—
456C3—121no—
457C3#8867611yescharacteristic, direct product factor
458C3C3×D3×#6711yescharacteristic
459C2—21no—
460C2—92no—
461C2—181no—
462C2—181no—
463C2—181no—
464C2C2×C3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC2×C3C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C23×C3
FrattiniC2×C3C2×C3×D32
FittingC22×C3×H(3,𝔽3)C22
socleC2×C32C2×D32
hypercenterC22×C3((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×C3×D4
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×C3×(C3⋊D3)⋊C3C2semidirect
2. C2×C3×((C3⋊D3)⋊C3)⋊C2C2semidirect
3. C3×#6235C2semidirect
4. C3×#638⋊C3C2semidirect
5. C3×#6200C2semidirect
6. C3×#6198C2semidirect
7. C3×((C3⋊D3)⋊C3)⋊C4C2semidirect
10. #88676C3direct (N × H)
11. C22×C3×H(3,𝔽3)C22semidirect
12. C2×C3×#272C22semidirect
15. C2×C3×(C3⋊D3)⋊C3C22semidirect
16. C2×C3×(C3⋊D3)⋊C3C22semidirect
18. C3×#1084C22semidirect
19. C3×(C3⋊Dic3)⋊C3C22semidirect
20. C22×C3×C3⋊D3D3semidirect
24. C3×#638D3semidirect
36. C22×(C3⋊D3)⋊C3C2×C3semidirect
37. C2×((C3⋊D3)⋊C3)⋊C2C2×C3semidirect
38. #6235C2×C3semidirect
40. #638⋊C3C2×C3semidirect
41. #6200C2×C3semidirect
42. #6198C2×C3semidirect
43. ((C3⋊D3)⋊C3)⋊C4C2×C3semidirect
49. C3×(C3⋊D3)⋊C3D4semidirect
56. C22×C33C2×D3semidirect
57. C22×C33C2×D3semidirect
60. C2×C3×C3⋊D3C2×D3semidirect
61. C2×C3×C3⋊D3C2×D3semidirect
71. C3×C3⋊Dic3C2×D3semidirect
79. C22×H(3,𝔽3)C22×C3semidirect
80. C2×#272C22×C3semidirect
87. C2×(C3⋊D3)⋊C3C22×C3semidirect
88. C2×(C3⋊D3)⋊C3C22×C3semidirect
90. #1084C22×C3semidirect
92. (C3⋊Dic3)⋊C3C22×C3semidirect
93. C3×H(3,𝔽3)C2×D4semidirect
95. C22×C3⋊D3C3×D3semidirect
107. #638C3×D3semidirect
144. C3×C3⋊D3#67semidirect
165. (C3⋊D3)⋊C3C3×D4semidirect
186. C22×C32C2×C3×D3semidirect
187. C22×C32C2×C3×D3semidirect
190. C2×C3⋊D3C2×C3×D3semidirect
191. C2×C3⋊D3C2×C3×D3semidirect
224. C3⋊Dic3C2×C3×D3semidirect
239. C33D3×D4semidirect
240. C33C2×#67semidirect
244. H(3,𝔽3)C2×C3×D4semidirect
309. C3⋊D3C3×#67semidirect
355. C22×C3((C3⋊D3)⋊C3)⋊C2semidirect
393. C32C3×D3×D4semidirect
394. C32C2×C3×#67semidirect
447. C22C3×((C3⋊D3)⋊C3)⋊C2semidirect
457. C3#88676direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×C3×(C3⋊D3)⋊C32
C2×C3×((C3⋊D3)⋊C3)⋊C22
C3×#62352
C3×#638⋊C32
C3×#62002
C3×#61982
C3×((C3⋊D3)⋊C3)⋊C42
C3×#38113
C3×D3×#673
#886763